Singularity Formation for the Compressible Euler Equations

Abstract

It is well-known that singularity will develop in finite time for hyperbolic conservation laws from initial nonlinear compression no matter how small and smooth the data are. Classical results, including Lax [14], John [13], Liu [22], Li-Zhou-Kong [16], confirm that when initial data are small smooth perturbations near constant states, blowup in gradient of solutions occurs in finite time if initial data contain any compression in some truly nonlinear characteristic field, under some structural conditions. A natural question is that: Will this picture keep true for large data problem of physical systems such as compressible Euler equations? One of the key issues is how to find an effective way to obtain sharp enough control on density lower bound, which is known to decay to zero as time goes to infinity for certain class of solutions. In this paper, we offer a simple way to characterize the decay of density lower bound in time, and therefore successfully classify the questions on singularity formation in compressible Euler equations. For isentropic flow, we offer a complete picture on the finite time singularity formation from smooth initial data away from vacuum, which is consistent with the small data theory. For adiabatic flow, we show a striking observation that initial weak compressions do not necessarily develop singularity in finite time. Furthermore, we follow [7] to introduce the critical strength of nonlinear compression, and prove that if the compression is stronger than this critical value, then singularity develops in finite time, and otherwise there are a class of initial data admitting global smooth solutions with maximum strength of compression equals to this critical value.

Singularity formation for the 1D compressible Euler equations with variable damping coefficient

“…For more general 2 × 2 strictly hyperbolic system including the 1D Euler equation, sufficient conditions for the derivative blow-up (the formation singularity) has been studied by many mathematicians (e.g. Lax [13], Zabusky [31], Klainerman and Majda [11], MacCamy and Mizel [17], Manfrin [16], Liu [15] and Cheng, Pan and Zhu [3]). In [13], Lax has established important formulas for solutions to 2 × 2 hyperbolic systems.…”

Optimal time-dependent lower bound on density for classical solutions of 1-D compressible Euler equations

“…For non-isentropic full Euler equations, before this paper, the only polynomial order upper bound of τ (lower bound of ρ) for general classical solution was established by Pan, Zhu and the author in [4]. More precisely, we showed density has a lower bound in the order of O(1 + t) −4/(3−γ) when 1 < γ < 3.…”

“…The lower bounds of density achieved in this paper can give us more precise estimate of life span of classical solution than those achieved in [4] and motivate us in searching lower bound of density for BV solutions including shock waves, which is a major obstacle in establishing large BV existence theory for Euler equations. Another interesting result on a time-dependent density lower bound for isentropic Euler-Poisson equations can be found in [17] by E. Tadmor and D. Wei.…”

“…The proof of Theorem 2.3 also relies on the uniform constant upper bound of density established in [8] by R. Young, Q. Zhang and the author for classical solutions when total variation of initial entropy is finite. One direct application of Theorem 2.3 is that one can use (2.13) to improve the life-span estimate established in [4] when 1 < γ < 3 which depends on the timedependent lower bound of density. We leave this to the reader.…”

“…For rarefactive piecewise Lipschitz continuous solutions, for any γ > 1, L. Lin first proved that the density has lower bound in the order of O(1 + t) −1 in [13] by introducing a polygonal scheme. A breakthrough for general classical solutions happens in a recent paper [4], in which R. Pan, S. Zhu and the author found a lower bound of density in the order of O(1 + t) −4/(3−γ) when 1 < γ < 3. Using this result together with Lax's decomposition in [12], Pan, Zhu and the author proved that gradient blowup of u 1 The author thanks Helge Kristian Jenssen who first pointed out this result to him.…”